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Full text of "Analytical Mechanics"

MOTION OF A PARTICLE                        127
the bodies are given the same angular velocity about a vertical axis through the point of suspension the particles will lie in the same horizontal plane.
6.   If the masses in the preceding problem are equal how will the tensile force vary with the length of the strings?
7.   Supposing the earth to be spherical discuss the variation in the weight of a body due to the rotation of the earth about its axis.
8.   The moon describes a circular path around the earth once in every 27 days, 7 hours, and 43 minutes; find the acceleration at the center of the moon due to the attraction of the earth.   Take 240,000 miles for the radius of the moon's orbit.
9.   If the earth rotated fast enough to make the weights of bodies vanish at the equator show that the plumb line at any latitude would become parallel to the axis of the earth.
10.   In the preceding problem what would the length of the day be?
11.   How much would the weight of a body be increased at latitude 30 if the earth stopped rotating?
12.   A particle suspended from a fixedjppint by a string of length I is projected horizontally with a speed "\f\lg\ show that the string will become slack when the particle has risen to a height 11.
13.   How much should the outer rail of a railroad track be raised at a curve in order that there be no lateral pressure on the rails when a train makes the curve at the rate of a mile a minute?   The radius of the curve is 1500 feet and the distance between the tracks is 4 feet 8 inches.
14.   Prove that a locomotive will upset if it takes a curve with a speed
greater thany |^, on tracks the outer rails of which are not raised,
where g denotes the gravitational acceleration, r the radius of the curve, a the distance between the rails, and h the height of the center of mass of the locomotive above the tracks.
16. Show that if there is no lateral pressure on the outer rails, while a car takes a curve, the relation
tan0 =  gr
is satisfied, where 0 is the angle the floor of the car makes with the horizon, v is the speed of the car, and r the radius of the curve.
111. II. Bodies Falling from Great Distances.  When the distance from which a body falls is not negligible compared with the radius of the earth the gravitational acceleration cannot be considered as constant during the fall. Therefore